Someone forgot their code!

Number Theory Level 2

Marilyn has forgotten her four-digit pin code! Fortunately, she remembers that when the two-digit number formed by the last two digits is subtracted from the two-digit number formed by the first two digits the result is 9. Furthermore she remembers that the sum of the squares of the first and last digits minus the sum of the squares of the second and third digits is exactly the two-digit number formed by the last two digits. What is Marilyn's four-digit pin code?


The answer is 2718.

Magnus Seidenfaden by Magnus Seidenfaden , 19, Denmark

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1 solution

Let the digits of the number be a , b , c , d a, b, c, d . Then each of them is an integer between 0 0 and 9 9 inclusive. We have 10 a + b − 10 c − d = 9 10a+b-10c-d=9 and a 2 + d 2 − b 2 − c 2 = 10 c + d a^2+d^2-b^2-c^2=10c+d , or 10 ( a − c ) + ( b − d ) = 9 10(a-c)+(b-d) =9 and ( a + c ) ( a − c ) + ( d + b ) ( d − b ) = 10 c + d (a+c)(a-c) +(d+b)(d-b) =10c+d . Therefore a + b = 9 c a+b=9c , and hence b = 8 c − 1 b=8c-1 . Therefore a = 2 , b = 7 , c = 1 , d = 8 a=2, b=7, c=1, d=8 , and the required number is 2718 2718

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